I haven’t seen any client that has a share as DM option.

Generally DMs on nostr are only semi-private right now. Nobody can read the message but metadata is leaked, I.e. the fact that a message has been sent (when / to whom) is public.

Better DM implementations are in the works. 

 This used to be the case but most clients have moved past NIP-04 now. There are some really excellent options on nostr now like https://www.0xchat.com 

 But this is a separate app. There isn't a way to do it in a client...yet? 

So either I have to copy paste my note ID into direct message in the client or come out of the client and go into a messaging app to send it. 

Can I hope that soon integration will come? 

 Are you on Android? I use this all the time on #Amethyst. I'll copy the note ID using the three dots at the top of the post then just past that into Amethyst DMs a lot of times. Amethyst supports gift wrapped messaging.  

 I am not (shame shame) but I can do this on the other clients. Not quite a share to DM button though!